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The Addition Rule on a Tree Diagram

(KS3, Year 7)

A tree diagram shows all the possible outcomes of an event and their probabilities. We can find the probability of one event or another event.

A Single Coin Toss

A single coin toss can be shown on a tree diagram. coin_toss_tree

The probability of each event can be read from each branch:

The probability of getting Heads is ¹⁄₂.
The probability of getting Tails is ¹⁄₂.

Imagine we wanted to find the probability of getting Heads or Tails. coin_toss_tree_addition

The probability of Heads or Tails is found by adding their probabilities.

¹⁄₂ + ¹⁄₂ = 1

How to Find the Probability of an Event or Another Event on a Tree Diagram

The addition rule can be used for more complicated tree diagrams. The tree diagram below is for a double coin toss. coin_toss_twice_tree

Question

What is the probability of both tosses landing the same side up?

To find the probability of an event or another event, add across the branches.

Step-by-Step:

1

Find the event given in the question. The event is the coin landing the same side up on both tosses. There are two ways this can happen:

Heads and Heads
Tails and Tails

We are finding the probability of Heads and Heads or Tails and Tails. These are the top branches and the bottom branches of the tree diagram.

addition rule tree diagram step 1

2

Use the multiplication rule to find the probabilities of the outcomes. Multiply the probabilities along the branches.

addition rule tree diagram step 2

For Heads and Heads:

P(Heads and Heads) = ¹⁄₂ × ¹⁄₂ = ¹⁄₄

For Tails and Tails:

P(Tails and Tails) = ¹⁄₂ × ¹⁄₂ = ¹⁄₄

3

Add the probabilities for these outcomes.

addition rule tree diagram step 3

¹⁄₄ × ¹⁄₄ = ²⁄₄

4

Simplify the fraction if possible. (The fraction in our example is already as simple as possible). Simplify the fraction by dividing the top and bottom numbers by their greatest common factor (which is 2).

2 ÷ 2 = 1 4 ÷ 2 = 2

Answer:

The probability of getting Heads and Heads or Tails and Tails is ¹⁄₂. We can also express this as a number (0.5) or a percentage (50%).

Lesson Slides

The slider below another real example of using the addition rule on a tree diagram.

What Is Probability?

A probability is a measure of how likely (how probable) an event is to happen. A probability is expressed as a number between 0 (impossible) and 1 (certain). The formula for finding a probability is shown below: probability_formula_mini

Top Tip

A Useful Check

There is a useful way to check that the probabilities on a tree diagram are all correct. The probabilities of each final outcome add up to 1: outcome_add_to_1